Date of Thesis

Spring 2024

Description

Given a circle inscribed in a polygon inscribed in the unit circle, if one connects all the vertices with line segments we get a family of circles called a package of Poncelet circles, due to its connection to a theorem of Poncelet. We are interested in where the centers of the Poncelet circles can be. Specifically, we have shown that if one of the circles in the Poncelet package is centered at 0, then all of the circles must be centered at 0 as well. This was proven by Spitkovsky and Wegert in 2021 using elliptic integrals but we provide a simple, geometric, and intuitive proof. These circles also connect to an object in operator theory called the numerical range. In 2016, Gau, Wang, and Wu asked the still open question, when is the numerical range of a partial isometry a circular disk? They showed that if the numerical range of an n×n matrix that is a partial isometry (i.e., preserves distance on the orthogonal complement of its kernel) is a circular disk, then the circle must be centered at 0 for n ≤ 4. Our work re-proves and simplifies their proof.

Keywords

Poncelet's Theorem, Packages of Poncelet Circles, Blaschke Products, Numerical Ranges, Partial Isometries

Access Type

Honors Thesis

Degree Type

Bachelor of Science

Major

Mathematics

Minor, Emphasis, or Concentration

Dance

First Advisor

Pamela Gorkin

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Included in

Analysis Commons

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